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<div class="titlepage"><div><div><h5 class="title">
<a name="boost_test.testing_tools.extended_comparison.floating_point.floating_points_comparison_impl"></a><a class="link" href="floating_points_comparison_impl.html" title="Tolerance-based comparisons">Tolerance-based
          comparisons</a>
</h5></div></div></div>
<p>
            Assertions in the <span class="emphasis"><em>Unit Test Framework</em></span> use two kinds
            of comparison. For <code class="computeroutput"><span class="identifier">u</span></code>
            being close to zero with absolute tolerance <code class="computeroutput"><span class="identifier">eps</span></code>:
          </p>
<pre class="programlisting"><span class="identifier">abs</span><span class="special">(</span><span class="identifier">u</span><span class="special">)</span> <span class="special">&lt;=</span> <span class="identifier">eps</span><span class="special">;</span> <span class="comment">// (abs)</span>
</pre>
<p>
            For <code class="computeroutput"><span class="identifier">u</span></code> and <code class="computeroutput"><span class="identifier">v</span></code> being close with relative tolerance
            <code class="computeroutput"><span class="identifier">eps</span></code>:
          </p>
<pre class="programlisting">   <span class="identifier">abs</span><span class="special">(</span><span class="identifier">u</span> <span class="special">-</span> <span class="identifier">v</span><span class="special">)/</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">u</span><span class="special">)</span> <span class="special">&lt;=</span> <span class="identifier">eps</span>
<span class="special">&amp;&amp;</span> <span class="identifier">abs</span><span class="special">(</span><span class="identifier">u</span> <span class="special">-</span> <span class="identifier">v</span><span class="special">)/</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">v</span><span class="special">)</span> <span class="special">&lt;=</span> <span class="identifier">eps</span><span class="special">;</span> <span class="comment">// (rel)</span>
</pre>
<p>
            For rationale for choosing these formulae, see section <a class="link" href="floating_points_comparison_theory.html" title="Theory behind floating point comparisons">Floating
            point comparison algorithms</a>.
          </p>
<p>
            Assertion <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_test_universal_macro.html" title="BOOST_TEST"><code class="computeroutput"><span class="identifier">BOOST_TEST</span></code></a> (when comparing floating-point
            numbers) uses the following algorithm:
          </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
                When either value <code class="computeroutput"><span class="identifier">u</span></code>
                or <code class="computeroutput"><span class="identifier">v</span></code> is zero, evaluates
                formula (abs) on the other value.
              </li>
<li class="listitem">
                When the specified tolerance is zero, performs direct (native) comparison
                between <code class="computeroutput"><span class="identifier">u</span></code> and <code class="computeroutput"><span class="identifier">v</span></code>.
              </li>
<li class="listitem">
                Otherwise, performs formula (rel) on <code class="computeroutput"><span class="identifier">u</span></code>
                and <code class="computeroutput"><span class="identifier">v</span></code>.
              </li>
</ul></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
              Therefore in order to check if a number is close to zero with tolerance,
              you need to type:
            </p>
<pre class="programlisting"><span class="identifier">BOOST_TEST</span><span class="special">(</span><span class="identifier">v</span> <span class="special">==</span> <span class="identifier">T</span><span class="special">(</span><span class="number">0</span><span class="special">),</span> <span class="identifier">tt</span><span class="special">::</span><span class="identifier">tolerance</span><span class="special">(</span><span class="identifier">eps</span><span class="special">));</span>
</pre>
</td></tr>
</table></div>
<p>
            The compatibility assertions <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_level_close.html" title="BOOST_&lt;level&gt;_CLOSE"><code class="computeroutput"><span class="identifier">BOOST_</span><span class="special">&lt;</span><span class="identifier">level</span><span class="special">&gt;</span><span class="identifier">_CLOSE</span></code></a> and <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_level_close_fraction.html" title="BOOST_&lt;level&gt;_CLOSE_FRACTION"><code class="computeroutput"><span class="identifier">BOOST_</span><span class="special">&lt;</span><span class="identifier">level</span><span class="special">&gt;</span><span class="identifier">_CLOSE_FRACTION</span></code></a> perform formula
            (rel).
          </p>
<p>
            The compatibility assertion <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_level_small.html" title="BOOST_&lt;level&gt;_SMALL"><code class="computeroutput"><span class="identifier">BOOST_</span><span class="special">&lt;</span><span class="identifier">level</span><span class="special">&gt;</span><span class="identifier">_SMALL</span></code></a> performs formula (abs).
          </p>
<p>
            The <span class="emphasis"><em>Unit Test Framework</em></span> also provides unary predicate
            <code class="computeroutput"><a class="link" href="../../../../boost/math/fpc/small_with_tolerance.html" title="Class template small_with_tolerance">small_with_tolerance</a></code> and
            binary predicate predicate <code class="computeroutput"><a class="link" href="../../../../boost/math/fpc/close_at_tolerance.html" title="Class template close_at_tolerance">close_at_tolerance</a></code> that
            implement formula (abs) and (rel) respectively.
          </p>
<h4>
<a name="boost_test.testing_tools.extended_comparison.floating_point.floating_points_comparison_impl.h0"></a>
            <span class="phrase"><a name="boost_test.testing_tools.extended_comparison.floating_point.floating_points_comparison_impl.tolerance_in_operator"></a></span><a class="link" href="floating_points_comparison_impl.html#boost_test.testing_tools.extended_comparison.floating_point.floating_points_comparison_impl.tolerance_in_operator">Tolerance
            in <code class="computeroutput"><span class="keyword">operator</span><span class="special">&lt;</span></code></a>
          </h4>
<p>
            Tolerance-based computations also apply to <code class="computeroutput"><span class="keyword">operator</span><span class="special">&lt;</span></code> and other relational operators. The
            semantics are defined as follows:
          </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
                <span class="emphasis"><em>less-at-tolerance</em></span> &lt;==&gt; <span class="emphasis"><em>strictly-less</em></span>
                and not <span class="emphasis"><em>close-at-tolerance</em></span>
              </li>
<li class="listitem">
                <span class="emphasis"><em>greater-at-tolerance</em></span> &lt;==&gt; <span class="emphasis"><em>strictly-greater</em></span>
                and not <span class="emphasis"><em>close-at-tolerance</em></span>
              </li>
<li class="listitem">
                <span class="emphasis"><em>less-or-equal-at-tolerance</em></span> &lt;==&gt; <span class="emphasis"><em>strictly-less</em></span>
                or <span class="emphasis"><em>close-at-tolerance</em></span>
              </li>
<li class="listitem">
                <span class="emphasis"><em>greater-or-equal-at-tolerance</em></span> &lt;==&gt; <span class="emphasis"><em>strictly-greater</em></span>
                or <span class="emphasis"><em>close-at-tolerance</em></span>
              </li>
</ul></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
              This implies that the exactly one of these: <code class="computeroutput"><span class="identifier">u</span>
              <span class="special">&lt;</span> <span class="identifier">v</span></code>,
              <code class="computeroutput"><span class="identifier">u</span> <span class="special">==</span>
              <span class="identifier">v</span></code>, <code class="computeroutput"><span class="identifier">u</span>
              <span class="special">&gt;</span> <span class="identifier">v</span></code>,
              passes with <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_test_universal_macro.html" title="BOOST_TEST"><code class="computeroutput"><span class="identifier">BOOST_TEST</span></code></a> at any given tolerance.
            </p></td></tr>
</table></div>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
              Relation <span class="emphasis"><em>less-at-tolerance</em></span> is not a <span class="emphasis"><em>Strict
              Weak Ordering</em></span> as it lacks the <span class="emphasis"><em>transitivity of
              the equivalence</em></span>; using it as predicate in <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">map</span></code> or any order-based STL algorithm
              would result in undefined behavior.
            </p></td></tr>
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